Earth's rotation and gravitation

[quasi426]

I don't understand why the physics book defines the centripetal force and gravitational force acting in different directions.

N - m*(acceleration of gravity) = -m*r*w^2

N = m*(acceleration of gravity) - m*r*w^2

Why don't the acceleration of gravity and centripetal acceleration add up? I would think that the normal force would be greater near the equator since there is both gravitational and centripetal forces in the same direction. But the book says the opposite. Thanks for the help.

 

[Doc Al]

Centripetal force is not a kind of force, it is just the name given to any force that pulls towards the center of some rotating system. (Centripetal just indicates the direction of the force, not the source of the force. It's similar to saying that a force acts horizontally or vertically. Centripetal means "towards the center".)

In the case of an object at the equator, the net centripetal force is just the net force acting towards the center. If the Earth didn't rotate, then there would be zero acceleration and N = mg. But the Earth does rotate, so there must be a net force acting centripetally on the object. Thus the normal force is less. If the Earth starting spinning faster and faster, at some point the object would be thrown off--the normal force would go to zero.

Looked at from the noninertial frame of the rotating Earth you can say that there is a centrifugal acceleration on the object that acts to pull the object away from the center. (Centrifugal means "away from the center".) So you could say that the acceleration due to gravity and the centrifugal acceleration add up, but they act in opposite directions.